| People have studied binomial coefficients for more than700years. The sequences of sums of powers of binomial coefficients have many very good properties which play important roles in lots of discussions and research about mathematical problems. For example, irrational number is proved and prime number is found by using the properties. In1978, Apery proved the irrationality of ζ(2) and ζ(3) using the properties. In2004, Agrawal, Saxena and Kayal gave the primality test, which is closely related to the congruence properties ofIn this paper, I will study some congruence properties of the sequences by using the theory and method of exact division and congruence and Diophantine analysis. It has four chapters, principally as follows:Firstly, the history background, meanings of study and main conclusions.Secondly, some basic theories, main theorems and lemmas.Thirdly, the congruence properties of modulo p, p2are proved.Fourthly, the congruence properties of modulo p are discussed. |
